Space of modular forms of level 255, weight 2, and trivial character

• Label: 255.2.a
• Level: $255$
• Weight: $2$
• Character orbit: 255.a
• Rep. character: $\chi_{255}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 255 = 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	255.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(72\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(255))\).

	Total	New	Old
Modular forms	40	11	29
Cusp forms	33	11	22
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(11\)

Trace form

11 * q + 5 * q^2 + 3 * q^3 + 17 * q^4 - q^5 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 11 * q^9 + q^10 + 12 * q^11 + 5 * q^12 + 2 * q^13 - 8 * q^14 - q^15 + 17 * q^16 - q^17 + 5 * q^18 - 4 * q^19 - 7 * q^20 + 8 * q^21 - 12 * q^22 + 8 * q^23 - 15 * q^24 + 11 * q^25 - 18 * q^26 + 3 * q^27 - 8 * q^28 + 2 * q^29 - 3 * q^30 - 8 * q^31 - 7 * q^32 - 12 * q^33 - 3 * q^34 + 17 * q^36 + 26 * q^37 - 28 * q^38 + 2 * q^39 - 3 * q^40 + 6 * q^41 - 24 * q^42 - 4 * q^43 - 12 * q^44 - q^45 - 32 * q^46 - 16 * q^47 - 3 * q^48 + 27 * q^49 + 5 * q^50 - q^51 - 58 * q^52 + 10 * q^53 - 3 * q^54 - 12 * q^55 - 72 * q^56 + 28 * q^57 - 34 * q^58 - 20 * q^59 - 7 * q^60 - 14 * q^61 - 64 * q^62 + 8 * q^63 - 19 * q^64 + 18 * q^65 - 8 * q^66 - 4 * q^67 - 7 * q^68 - 16 * q^69 - 12 * q^70 + 9 * q^72 + 38 * q^73 - 10 * q^74 + 3 * q^75 - 8 * q^76 + 6 * q^78 - 16 * q^79 + q^80 + 11 * q^81 - 6 * q^82 - 4 * q^83 - 4 * q^84 + 3 * q^85 + 44 * q^86 - 6 * q^87 - 12 * q^88 + 38 * q^89 + q^90 - 8 * q^91 + 56 * q^92 + 24 * q^93 + 20 * q^94 - 20 * q^95 - 23 * q^96 + 30 * q^97 + 61 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(255))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	17
255.2.a.a	$255$	$2$	255.a	1.a	$1$	$2$	$2$	$2.036$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	255.2.a.a	$1$	$0$	\(1\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
255.2.a.b	$255$	$2$	255.a	1.a	$1$	$2$	$2$	$2.036$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	255.2.a.b	$1$	$0$	\(3\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
255.2.a.c	$255$	$2$	255.a	1.a	$1$	$3$	$3$	$2.036$	3.3.229.1	None	✓	✓	✓	✓	255.2.a.c	$1$	$0$	\(0\)	\(3\)	\(3\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
255.2.a.d	$255$	$2$	255.a	1.a	$1$	$4$	$4$	$2.036$	4.4.13768.1	None	✓	✓	✓	✓	255.2.a.d	$1$	$0$	\(1\)	\(4\)	\(-4\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+q^{3}+(2-\beta _{1})q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(255))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(255)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)